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In this paper, a new multilevel correction scheme is proposed to solve Stokes eigenvalue problems by the finite element method. This new scheme contains a series of correction steps, and the accuracy of eigenpair approximation can be improved after each step. In each correction step, we only need to solve a Stokes problem on the corresponding fine finite element space and a Stokes eigenvalue problem on the coarsest finite element space. This correction scheme can improve the efficiency of solving Stokes eigenvalue problems by the finite element method. As applications of this multilevel correction method, a multigrid method and an adaptive finite element technique are introduced for Stokes eigenvalue problems. Some numerical results are given to validate our schemes. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present an a posteriori error estimator and the lower bound for a nonconforming finite element approximation, i.e. the extended Crouzeix–Raviart element, of the Laplace eigenvalue problem. Under the guideline of the analysis to the Laplace source problem, we first give out an error indicator and prove it as the global upper and local lower bounds of the approximation error. We also give the lower-bound analysis for this type of nonconforming element on the adaptive meshes. Some numerical experiments are presented to verify our theoretical results. [Copyright &y& Elsevier]

Rannacher discovered by numerical results that the Morley element eigenvalues could approximate the exact eigenvalues from below. This discovery is very important in engineering and mechanics computing. This note provides a theoretical proof for Rannacher's observations. [ABSTRACT FROM AUTHOR]

Abstract: The paper provides a general procedure or method to produce asymptotic error expansion for the eigenvalue approximations of second order elliptic problems by the mixed finite element method. We obtain a transform lemma for the error of the eigenvalue approximations. As an application of the transform lemma, the asymptotic error expansion of the eigenvalue approximations for the second order elliptic problem by the lowest order Raviart–Thomas mixed finite element method is given by means of integral identity technique. Based on such an error expansion, Richardson extrapolation technique is applied to improve the accuracy of the eigenvalue approximations. [Copyright &y& Elsevier]

Abstract: The paper explores new expansions of eigenvalues for in with Dirichlet boundary conditions by Wilson’s element. The expansions indicate that Wilson’s element provides lower bounds of the eigenvalues. By the extrapolation or the splitting extrapolation, the convergence rate can be obtained, where is the maximal boundary length of uniform rectangles. Numerical experiments are carried to verify the theoretical analysis made. It is worth pointing out that these results are new, compared with the recent book, Lin and Lin [Q. Lin, J. Lin, Finite Element Methods; Accuracy and Improvement, Science Press, Beijing, 2006]. [Copyright &y& Elsevier]

Abstract: The paper provides new expansions of leading eigenvalues for in S with the Dirichlet boundary condition on by finite elements, with the support of numerical experiments. The theoretical proof of new expansions of leading eigenvalues is given only for the bilinear element . However, such a new proof technique can be applied to other elements, conforming and nonconforming. The new error expansions are reported for the elements and other three nonconforming elements, the rotated bilinear element (denoted by ), the extension of (denoted by ) and Wilson''s element. The expansions imply that and yield upper bounds of the eigenvalues, and that and Wilson''s elements yield lower bounds of the eigenvalues. By the extrapolation, the convergence rate can be obtained, where h is the boundary length of uniform rectangles. [Copyright &y& Elsevier]